Suppose that AB and C are sets and gA B and f BC are surject

Suppose that A,B and C are sets, and g:A--> B and f: B-->C are surjective functions. Prove that fg: -->C is also surjective. please explain each step

Solution

Suppose that f o g is surjective. Let z belongs to C. Then since f o g is surjective, there exists x belongs to A, such that (fg)(x) =f(g(x)) =z. Therefore if we let y=g(x) belongs to B, then f(y) =z. Thus f is surjective.

Therefore, C is also surjective.